I'm not an Ubuntu user, so I don't really know how you would upgrade your software on that platform -- this is where "upgrade" replaces your existing out-of-date software installation.
The software you did download, a "dot-run" file, can be executed with
su - or sudo
<enter root password if prompted>
which may -- may! -- upgrade (actually, I suspect that it will, but I don't know diddly about the strange and wondrous workings of Ubuntu).
... and, I know even less about Mint (looking at the Ubuntu logo on your posts, eh?). Suspect, though, that you can do the above.
Generally, HP provides full support for every product they make in Linux so doing the above ought to (but may not!) get you where you want to go. You might want to check with folks that are familiar with Ubuntu before you do, but, on the other hand, what the heck. Up to you. I would also think that the current version would be available at Ubuntu's upgrade site (Yum or something like that? Dunno).
When you do get 3.12.2 installed (or upgraded), you can run hp-setup
as "you" (you will be prompted to enter the root password as you go) and hp-setup
will make all the connections you need in CUPS for you.
Hope this helps some.